We sample the latent cases \(S_n^N\)  along with their associated age groups \(A_n^N\)  falling into the subinterval \(\mathcal {T}_n\)  by applying Algorithm REF , which is a simulation procedure based on the branching structure of the Hawkes process [1]}. The proposed algorithm is a superposition of Poisson processes in the interval \(\mathcal {T}_n\) ; the descendants of each latent event at \(t_i\)  form an inhomogeneous Poisson process with intensity
\(\lambda _i(t)=h(t-t_i)\sum \limits _a\gamma _{na}\frac{S_{t,a}}{N_a}m_{aa_i}\) 
